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Hyperbolic angles in Lorentzian length spaces and timelike curvature bounds

Tobias Beran, Clemens Sämann

TL;DR

The paper introduces hyperbolic angles in Lorentzian length spaces and uses them to characterize timelike curvature bounds via an angle monotonicity principle. By defining the timelike tangent cone, exponential/logarithmic maps, and a Lorentzian law of cosines, it builds a cohesive framework linking angle behavior to synthetic curvature via $K$-monotonicity and model-space comparisons. The main contribution is proving that timelike curvature bounds are equivalent to a monotonicity condition on comparison angles, which also enables a robust non-branching result for timelike geodesics and provides tools for future globalization and rigidity results in Lorentzian geometry. These developments enhance the study of spacetimes with low regularity and have potential implications for singularity theory, causality, and general relativity in synthetic settings, including possible extensions to Lorentzian analogues of classical comparison theorems. All mathematical notation is presented with explicit delimiters to support integration into knowledge bases and search pipelines.

Abstract

Within the synthetic-geometric framework of Lorentzian (pre-)length spaces developed in Kunzinger and Sämann (Ann. Glob. Anal. Geom. 54(3):399--447, 2018) we introduce a notion of a hyperbolic angle, an angle between timelike curves and related concepts like timelike tangent cone and exponential map. This provides valuable technical tools for the further development of the theory and paves the way for the main result of the article, which is the characterization of timelike curvature bounds (defined via triangle comparison) with an angle monotonicity condition. Further, we improve on a geodesic non-branching result for spaces with timelike curvature bounded below.

Hyperbolic angles in Lorentzian length spaces and timelike curvature bounds

TL;DR

The paper introduces hyperbolic angles in Lorentzian length spaces and uses them to characterize timelike curvature bounds via an angle monotonicity principle. By defining the timelike tangent cone, exponential/logarithmic maps, and a Lorentzian law of cosines, it builds a cohesive framework linking angle behavior to synthetic curvature via -monotonicity and model-space comparisons. The main contribution is proving that timelike curvature bounds are equivalent to a monotonicity condition on comparison angles, which also enables a robust non-branching result for timelike geodesics and provides tools for future globalization and rigidity results in Lorentzian geometry. These developments enhance the study of spacetimes with low regularity and have potential implications for singularity theory, causality, and general relativity in synthetic settings, including possible extensions to Lorentzian analogues of classical comparison theorems. All mathematical notation is presented with explicit delimiters to support integration into knowledge bases and search pipelines.

Abstract

Within the synthetic-geometric framework of Lorentzian (pre-)length spaces developed in Kunzinger and Sämann (Ann. Glob. Anal. Geom. 54(3):399--447, 2018) we introduce a notion of a hyperbolic angle, an angle between timelike curves and related concepts like timelike tangent cone and exponential map. This provides valuable technical tools for the further development of the theory and paves the way for the main result of the article, which is the characterization of timelike curvature bounds (defined via triangle comparison) with an angle monotonicity condition. Further, we improve on a geodesic non-branching result for spaces with timelike curvature bounded below.
Paper Structure (15 sections, 40 theorems, 110 equations, 15 figures)

This paper contains 15 sections, 40 theorems, 110 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Main results and outline of the article
  3. Notation and conventions
  4. A brief introduction to Lorentzian (pre-)length spaces
  5. Curvature comparison for Lorentzian pre-length space s
  6. Comparison (model) spaces
  7. Triangle comparison
  8. Angles
  9. Definition and basic properties
  10. Angles between timelike curves of the same time orientation
  11. Timelike tangent cones
  12. Exponential and logarithmic map
  13. Angles between timelike curves of arbitrary time orientation
  14. Monotonicity comparison
  15. Law of Cosines

Key Result

Theorem 3.1

Let $(X,d,\ll,\leq,\tau)$ be a strongly causal and locally causally closed Lorentzian pre-length space with $\tau$ locally finite-valued and locally continuous. Let $\alpha,\beta,\gamma\colon[0,B)\rightarrow X$ be timelike curves with coinciding time orientation starting at $x:=\alpha(0)=\beta(0)=\g

Figures (15)

  • Figure 1: The three signed angles of a triangle.
  • Figure 2: The $\omega=0$ angle with $\sigma=1$ appears large to our Euclidean intuition.
  • Figure 3: The three one-sided comparison situations (case (3) consists of two cases: one can distinguish whether $x$ points to the future or the past).
  • Figure 4: On the left, the configuration in $X$ is shown. The grey area is $I(a,c)$, the red line is $\beta([s_-,s_+])$. In the comparison picture on the right, two points are connected by a line if this line has the same length as the corresponding line in $X$. The dashed line need not have the same length.
  • Figure 5: As the red lines are null and the dashed lines are timelike, we deduce that $\bar{b}_{s_-}$ is below $L_{s_-}$ and $\bar{b}_{s_+}$ is above $L_{s_+}$.
  • ...and 10 more figures

Theorems & Definitions (121)

  • Theorem 3.1: Triangle inequality for (upper) angles
  • Theorem 3.10: Triangle comparison implies future $K$-monotonicity comparison
  • Theorem 4.8: Timelike non-branching
  • Theorem 4.13: Equivalence of triangle and monotonicity comparison
  • Definition 1.1: Sectional curvature
  • Definition 1.2: Causal space
  • Definition 1.3: Lorentzian pre-length space
  • Definition 1.4: Causal, timelike and null curves
  • Definition 1.5: Length of curves
  • Definition 1.6: Intrinsic space
  • ...and 111 more